Electrical power computer apparatus



Jan. 7, 1964 1.. K. KIRCHMAYER 3,117,221

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ATTORNEY Jan, 7, 1964 L. K. KIRCHMAYER 3,117,221

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ELECTRICAL POWER COMPUTER APPARATUS Filed April 30, 1959 8 Sheets-Sheet 5 Jan. 7, 1964 L. K. KIRCHMAYER ELECTRICAL POWER COMPUTER APPARATUS 8 Sheets-Sheet 6 Filed April 30, 1959 R N QQN QQN Q...N4 s a A mu la R. Y. M m E w? a" w mw MQN 1 WQQQNEOU Ill 1 [I Jan. 7, 1964 L. K. KIRCHMAYER ELECTRICAL. POWER COMPUTER APPARATUS 8 Sheets-Sheet 7 Filed April 30, 1959 wwwm www 2. m m TA R MW m Nc a 1m A M (W mmw uo\ wmw 1964 L. K. KIRCHMAYER 3,117,221

ELECTRICAL POWER COMPUTER APPARATUS Filed April 30, 1959 a Sheets-Sheet a INCREMENML cosr X1 MICEEMENTAL C 05 T c x3 /NCEfME/VTAL 605 T s-x4 lNCEEME N TAL COST IN VEN TOR. LEO/V K. K/ECHMA YER.

ATTORAE'X 3,117,221 ELEiZTRICAL PGWER CGMPUTER APiARATUS Leon K. Kirchrna er, Scotia, N.Y., assignor to General Electric Company, a corporation of New York Filed Apr. 30, $59, Ser. No. 81%,(362 l tllaims. (Cl. 235-185) This invention relates to apparatus for computing the economic interchange of power between member systems of a power pool and more particularly to apparatus for computing the optimum amount of power to be delivered to each member system of a power pool by the other member systems while also computing the economic allocation of generation within each member system.

This application is a continuation-in-part of U.S. patent application 792,728, filed February 12, 1959, now abandoned.

The rapidly increasing demand for electrical energy has stimulated both an increase in the number of stations within power systems and the expansion of transmission networks for interconnecting many power systems. As stations hav increased and networks have expanded, the problems of maintaining cflicicnt power system operation and power interchange schedules have also increased.

Within each power system, it is no longer economically feasible to generate power at the side of the load. Instead, a modern power system comprises many generators and generating stations interconnected by a system power transmission network to transmit power to various load sites th oughout the system. However, a penalty in the form of transmi sion line power losses is paid for transmission of power between geographically remote generators and loads. In maintaining an efliciently operating power system, these power transmission losses cannot be neglected; and, therefore, power companies today seek to improve their operating economy by including the effects of transmission losses when scheduling generator and generating station power allocations. 1

It is known that when a plurality of generating units are interconnected to supply a system load, maximum economy is achieved if the incremental cost of power received at the system load is the same for all generating units. For example, see L. K. Kirchmayer, Economic Operation of Power Systems, chapter 5, John Wiley & Sons, Inc, New York, 1958. When the effects of transmission losses are included in preparing a system generation schedule for controlling the power to be supplied by each of the generating units so as to deliver the required power to the system load at a common incremental cost, the complex task normally prevents the dispatcher from manually maintaining the generation schedule current with changes in the system load. For example, most of the generators will deliver a particular Value of power at different incremental costs, and this incremental cost for each generator increases as the generated power increases. Furthermore, a continuous determination must be made as to whether it is cheaper to send high-cost power a short distance or low-cost power a long distance. Therefore, to aid the dispatcher, computing systerns are now available for computing the proper generation of each generating unit to satisfy system load requirements and to maintain a common incremental cost of power received at the system load. Such computing systems are described in US. Patent 2,839,692, issued in the name of L. K. Kirchmayer, and assigned to the assignee of the present invention.

Although the prior art contains descriptions of computing systems for determining the economic operation of generating units within a local power transmission system, it has been the practice heretofore to have neighboring power systems interchange power according to predetermined schedules. Thus, the computing systems of the i ite States Patent Q aforementioned patent compute the economic generation schedule of the local system generating units so as to maintain the interchange power from other local systems at a preselected value.

A power transmission system wherein a plurality of local systems are interconnected to exchange power is hereinafter termed an integrated power transmission system. Power may be transferred between a pair of local s stems of an integrated power transmission system by direct transmission over tie lines connecting together the two local systems, by wheeling power through a third local system which serves to connect tog her the two local systems, or by a combination of these two transmission methods. Such an integrated power transmission system is most economically operated only when each local system can receive power from each of the other local systems thereof at the same incremental cost, and when this incremental cost is the same as the incremental cost of power received from the local generating units. in preparing an interchange schedule for an integrated power transmission system, consideration must also be given as to whether it is more economical for each local system to receive or to supply interchange power, recognizing the transmission losses involved. This schedule must also consider the common incremental cost of power of each local system and the effect that the receipt or supply of interchange power has on altering this common incremental cost. To prepare a schedule for determining the most economic generation of an integrated system is, therefore, a time-consuming task of great magnitude. If the schedule is computed manually, it must be prepared many hours ahead of its time of employment, and only predicted values of load can be economically satisfied. To maintain such a schedule current, so as to meet the actual and changing load conditions in the integrated system, is obviously impossible if only manual computation is to be employed. Therefore, an integrated power transmission system can only be kept continuously operating most economically by employing an automatic computing system, which will continuously consider the mass of data involved and continuously compute the required power generation schedule.

It is, therefore, the principal object of this invention to provide apparatus for automatically computing the power requirements of an integrated power transmission system.

Another object of this invention is to provide apparatus for automatically computing the interchange power requirements for most economical operation of an integrated power transmission system.

Another obiect of this invention is to provide apparatus for automatically computing the algebraic value of power received by each local power transmission system of an integrated power transmission system for most economical operation of the integrated system.

Another object of this invention is to provide apparatus for automatically computing the power generation of each local power transmission system of an integrated power transmission system for receipt of interchange power by each local system from the other local systems at a common incremental cost.

Another object of this invention is to provide apparatus for automatically computing the power generation of all generating units of an integrated power transmission system for receipt of power by each local system of said integrated system from the other local systems and from the local generating units at a common incremental cost.

Another object of this invention is to provide apparatus for automatically computing the interchange power requirements of an integrated power transmission system for most economical operation thereof, while also computing the corresponding economic allocation of generaion within each local power transmission system.

Another object of this invention is to provide apparatus for automatically computing the power transferred on each of a plurality of transmission paths which connect a local power transmission system with at least one other local power transmission system.

Another object of this invention is to provide apparatus for automatically computing the power transferred on each of the plurality of transmission paths which interconnect the local power transmission systems of an integrated power transmission system.

Another object of this invention is to provide apparatus for automatically computing the incremental cost of power received at a predetermined location in an integrated power transmission system from a local power transmission system thereof when at least a portion of said power has been transmitted to said predetermined location through another local power transmission system.

Another object of this invention is to provide apparatus for automatically computing the incremental cost of losses for transmitting power through a local power transmission system of an integrated power transmission system when transferring power from one location to another in said integrated system.

Another object of this invention is to provide apparatus for automatically computing the incremental cost of power exchanged between an integrated transmission system and a power transmission system external thereto.

The foregoing objects are achieved by providing apparatus for generating signal pairs representing the incremental costs of power at predetermined locations in an integrated power transmission system for exchange of power by pairs of the it local power transmission systems, and in response to differences between the two values of each signal pair, for computing the required change of generation of each of the local systems to equalize the values of the corresponding signal pair. A local computer of the type previously described is provided for computing for each local system the economic allocation of generation therein. Each of the local computers is responsive to a first signal representing the desired total power to be consumed in the corresponding local system and a second signal representing the algebraic value or" power delivered to the local system by the other local systems of the integrated power transmission system, for computing the required power generataion by each of the local power sources to satisfy the local system load requirements and to deliver power to a predetermined location in the local system at a common incremental cost. A computing means for each one of 11-1 different pairs of the local systems computes a pair of incremental costs of power at a corresponding predetermined location in the integrated system for a hypothetical exchange of an increment of power between the two systems of the pair, the computation of this cost pair being based on the two common incremental costs of the corresponding pair of local systems. A comparison means determines the difference between the values of each computed pair of incremental costs. In response to each of these differences, a signal generator adjusts the values of the corresponding afore mentioned second signals in such a sense as to tend to reduce the magnitude of the difference.

Additional computing means responds to signals representing the power delivered by the individual generating stations of all of the local systems and to the signals representing the algebraic value of power delivered to each of said local systems by the other local systems for computing the power transmitted on each of the individual transmission paths of the integrated system. Wherever the aforementioned incremental cost of power delivered to a predetermined point is affected by power wheeled through a local system, additional computing means is provided to determine the contribution of the wheeling losses to this incremental cost. This latter type of computing means is responsive to the common incremental cost of power of the local system through which the 4 power is wheeled and to the values of power transmitted on the transmission paths which transfer power into and out of this local system for computing the incremental cost of losses for wheeling power therethrough.

This invention will be described with reference to the accompanying drawings, wherein;

FIGURE 1 is a block diagram of a single-area computing apparatus useful in the embodiment of this invention;

FIGURE 2 is a schematic diagram of a computing matrix useful in the embodiment of this invention;

FIGURE 3 is a block diagram of a station unit useful in the local computer of FIG. 1;

FEGURE 4 is a map of the integrated power transmission system to be maintained in economic dispatch by the embodiment of this invention;

FIGURE 5 is a block diagram of the concept of operation of the embodiment of this invention;

FIGURES 6a and 6b constitute a block diagram of an embodiment of this invention;

FIGURE 7 is a schematic diagram of the power consumption determination network for employment with the embodiment of FIG. 6;

FIGURE 8 is a schematic diagram of the foreign power interchange controls for employment with the embodiment of FIG. 6; and

FIGURE 9 is a schematic diagram of the foreign power cost weighting network for employment with the embodiment of FIG. 6.

The integrated system computing apparatus of this invention determines the interchange power requirements of an integrated power transmission system for cost economical operation thereof, delivers signals representing the interchange power contribution required of each local power transmission system, and includes a local computer of the type described in the aforementioned Kirchmayer Patent 2,839,692 to compute for each local system, the economic allocation of generation therein. Local computers of this type are further described in chapter 6 of the aforementioned Economic Operation of Power Systems by L. K. Kirchmayer. Each local computer responds to a signal representing the desired total power to be consumed in the corresponding local system and a signal representing the required interchange power contribution of the local system, to calculate the required power generation of each power source of the local sys tern to satisfy the power consumption and interchange contribution requirements while delivering power to a predetermined location in the corrresponding local system at a common incremental cost. Alterations of the interchange power requirements of the system are continuously calculated by the integrated system computing appara tus in response to changes in the common incremental costs determined by the local computers.

For a better understanding of the computing apparatus of this invention, a local computer useful in the embodiment thereof will be described.

Local Computer Theory A local computer calculates the required power generation by each generating unit of the corresponding local power transmission system to satisfy the power consumption and interchange power contribution requirements thereof while delivering power to a predtermined location in the corresponding local system at a common incremental cost. T he total required generation by all of the local system generating units is given by E Gi R i' L I where P represents the power generated by the i generating unit of the local system, P represents the local system load requirements,

For most economical operation of the local system to satisfy the power consumption and interchange power contribution requirements, the total cost of operation, F of the local system (in dollars per hour) must be minimized. This total cost is a function of the power delivered by all local system generating units; i.e.,

Equation 1 represents a constraint on the independent variable determining the total cost of Equation 2, and may be rewritten as R+ L I"ZPGi= where represents the constraining equation.

It is known that where such a function, F of n variables which are related by a constraining equation, 1/, is to be minimized, the values of the variables which render this function a minimum may be determined by equating to zero the partial derivatives of the auxiliary function F t+M wherein A is regarded as a constant. This is shown, for example, in E. Goursat and R. Hedrick, Mathematical Analysis, vol. 1, Ginn and Company, New York, 1904, pp. l28 and 129.

Equating the partial derivatives of Equation 4 to zero there result,

tion for generating unit, 2' in terms of, for 631- ample, dol ars per megawatt-hour,

represents the ratio of change in transmission OPGi losses in the local system to change in power of generating unit 11 when delivering an increment of power from generating unit 2" to the hypothetical load center of the local system, and

A may be interpreted as the incremental cost of power received by the load center, in terms of dollars per megawatt-hour.

(The preceding equations and their interpretation may also be found in chapter of the aforementioned Economic Operation of Power Systems by L. K. Kirchmayer.)

Equations 6, which are termed the intra-area coordination equations for the local system, specify that, for optimum economy, the incremental cost of received power at the load center of the local power transmission system should be the same for all generating units. Equations 6 describe the general mode of operation of the local computer shown in FIG. 1.

The total transmission losses of a local power transmission system may be given by L ZZ mnm n m n where P P represent the source loadings the local system, such as the power supplied to the system load by generating units of the system and power transferred into the system on tie-lines, and

E represents constants determined by the self and mutual impedances of the local system transmission network and the pattern of the loads on the network.

PJZE w 5 -gpBmaPm p QEZP B (s where represents the ratio of change in the j tieline G1 power flow to change in power of the i generating station when deliveringan increment of power from this z' generating station to the hypothetical load center of the local system.

in the local computer of FIG. 1, the incremental loss terms of Equation 8 that involve the terms may be neglected since they are usually small compared with the terms retained. Therefore, Equation 8 may be simplified as OF L G Z2B P =X m Substituting Equation 8a in Equation 6, there result:

or, +k2B P i OF l t21s P i 0F HZ BMPEA Equations 9 may be transposed to 6F, gPE)\ ?nf,2BmGlP or t \);,2B 2P g; =A-AE2B P (10 Equations describe the specific mode of operation of the local computer shown in FIG. 1.

Local Computer The particular local computer of FIG. 1 is employed to compute the economic operation of a local power transmission system comprising five steam power generating stations, this local power transmission system being designated as the Southern (S) local power transmission system in the integrated power transmission to be hereinafter described. Each generating station comprises one or more generators. In the local computer of FIG. 1, station units 30, 31, 32, 33 and 34 are provided to represent respectively the five steam power generating stations of the Southern local system. Each of station units -34- delivers a signal representing the total power delivered by the corresponding generating station.

The output signal of station unit 3% is designated as P of station unit 31 as P etc. This designation identifies each of these signals as representing the power of a particular steam power generating station of the Southern local system. All of the output signals of station units 30414 are applied to a summing amplifier 36, the output signal thereof representing the total power de livered by all the steam power generating stations of the local power transmission system. Summing amplifiers are well known in the art and, therefore, summing amplifier 36 may be of any conventional type. The output signal of summing amplifier 36 is applied to a differential amplifier 37. Diiferential amplifiers are also well known in the .art and, therefore, differential amplifiers 37 may be of any conventional type. Examples of summing and differential amplifiers useful in the embodiment of this invention are described in G. A. Kern and T. M. Korn, Electronic Analog Computers, McGraw-Hill Book Company, Inc., New York, 1952. Of course, it is not necessary specifically to employ a difierential amplifier, but, instead, a summing amplifier may be provided with negative input signals to represent the quantities to be subtracted from applied quantities represented by positive input signals. However, for a clearer understanding of the invention the symbol of a differential amplifier is illustrated in FIG. 1.

A signal representing the desired total power to be consumed in the Southern system is applied to an input terminal 38, which is coupled to differential amplifier 37. The power consumption represented by the signal applied to terminal 38 includes the Southern system load requirements, designated P in Equation 1, and the Southern system transmission losses, designated P in Equation 1. A signal representing the algebraic value of power delivered to the Southern system by power sources external thereto is applied to an output terminal 39, which is coupled to differential amplifier 37. The signal applied to terminal 39 represents the interchange power contribution requireemnts of the Southern local system, designated P in Equation 1. If the polarity of the signal is positive, the local system receives the interchange power quantity represented thereby; and if the polarity of the signal is negative, the local system transmits the interchange power quantity represented thereby. Differential amplifier 37 delivers an output signal representing the difierence between the sum of the signals received from amplifier 36 and terminal 39 and the signal received from terminal 38. Thus, the output signal of diiferential amplifier 37 represents the central expression of constraining Equation 3. Whenever this central expression deviates from zero, so that Equation 3 is not satisfied, differential amplifier 37 provides a corresponding output signal of proper polarity. Thus, when the required power consumption by the Southern system differs from the power supplied thereto (by local generating stations and external power sources) a corresponding error or difference signal is delivered by differential amplifier 37. The error signal of differential amplifier 37 is applied to a servo amplifier 65, the output signal thereof being applied to -a servomotor 66.

The output signal of each of the station units 30-34 is applied to a respective row of a computing matrix 40. Also applied to respective rows of matrix 40' are signals representing the power transferred into the local system on the tie-line connected thereto. Computing matrix 40 is emploped to provide signals representing respectively the ratio of change in transmission losses in the Southern system to change in power of a generating station for all of the Southern system generating stations. In matrix 48 the values of the signals representing the generating station powers and the tie-line powers are multiplied by the appropriate B quantities and related products are summed in accordance with Equation 8. Therefore, the specific computing matrix of FIG. 1 is identified as an incremental transmission loss matrix in accordance with the output signals provided thereby.

Computing matrix 40 is shown in detail in FIG. 2. However, the matrix of FIG. 2 is useful for all computing matrices to be referred to hereinafter in this description. The computing matrix comprises a plurality of input rows, shown as horizontal leads, a plurality of output columns, shown as vertical leads, and a plurality of multiplying means, shown as otentiometers.

Each input signal is received at a respective input terminal 42 and applied directly to a corresponding positive lead 43. Each input signal is inverted by a corresponding inverting amplifier 44 and applied to the corresponding negative lead 45. A plurality of otentiometers 47 receive an input signal from a corresponding one of switches 48. Each one of potentiometers 47 and its corresponding switch 48 is preadjusted in accordance with the desired value of B to be represented. The values of B may be scaled to have magnitudes ranging from 0 to 1.0 and bear either a positive or a negative polarity designation. The magnitude of E is preadjusted by setting the position of the movable arm of the corresponding one of potentiometers 47. The corresponding polarity designation of B is preadjusted by setting the position of the corresponding one of switches 48. A switch 48 is set to receive a signal from the associated one of positive leads 43 if the polarity designation of the corresponding value of B is positive. A switch 48 is set to receive a signal from an associated negative lead 4-5 if the polarity designation of B is negative. Therefore, the association of signs positive and negative with leads 43 and 45 is not an indication of the polarity of the signal transferred on each lead, but is, instead, an indication of the polarity designations which 'may be provided by those switches 48 which are connected thereto.

A plurality of combining leads 5d are provided, the number of combining leads corresponding to the number of input signals delivered to the computing matrix. The specific drawing of FIG. 2 is shown merely for il lustrativc purposes and therefore does not show the compiete structure of a particular computing matrix. Thus, neither all of the positive and negative leads, nor all of the combining leads are shown, but, instead, only a sulficient number of these leads are illustrated to demonstrate the principle of operation and construction of a computing matrix. A potentiometer 47 for each positive and negative lead pair is coupled to each one of combin ing leads 5% The movable arm of each potentiometer transfers a signal through a corresponding resistor 52 to the corresponding one of combining loads 59. Each one of combining leads 5t) delivers the signals received thereby to a corresponding summing amplifier 54. Each summing amplifier 54 delivers an output signal representing the summation of the signals received by the corresponding combining lead :Eii from the plurality'of potentiometers 47 coupled thereto. A number of summing amplifiers 54 equal to the number of input signals representing the generating station powers are connected to corresponding ones of a plurality of output terminals 55. The output signals available at terminals 55 represent respectivey the incremental transmission losses; i.e., the rate of change in transmission losses in the local transmission system, as indicated in Equation 8a. Thus, for the Southern system and computing matrix 40, five output terminals 55 are provided to deliver the respective signals representing these incremental transmission losses for the live generating stations of the Southern system.

Consider specifically the generation of the incremental transmission loss value for the first generating station of the Southern system. Equation 8a specifically written for this computation is Inasmuch as there are five generating stations in the Southern system and six tie-lines are connected thereto, the summation of Equation 11 involves eleven terms. The eleven potentiometers 47 coupled to the corresponding combining lead are adjusted for the corresponding one of the eleven values of 3 The corresponding ones of switches are connected to the associated positive lead 43 or negative lead 45 in accordance with the polarity designation of the corresponding B Each input si nal is multiplied in the corresponding one of these eleven potentiometers 47 by the value of B represented thereby. The output signal of each of these eleven potentiometers represents the product of the particular value of E and the value of power represented by the input signal received thereby. The output signals of these eleven potentiometers are algebraically added by the corresponding one of summing amplifiers 54, the output signal of which corresponds to the incremental transmission loss represented by Equation 11. Each one of summing amplifiers 54 has a gain of 2 to account for the factor of 2 in Equation 11.

A number of additional summing amplifiers 54 and corresponding output terminals 56 equal to the number of tie-lines connected to the local system are also shown in FIG. 2. For the Southern system, six terminals 56 are provided. The significance of the signals delivered at terminals 56 will be described hereinafter.

Only that portion of computing matrix 40 provided for delivering signals to output terminals 55 is considered part of the local computer. The entire computing matrix 4'? is part of a single-area computing apparatus which comprises all elements of PEG. 1, and which will be described hereinafter. In FIG. 1, the portion of computting matrix 4a? to the left of the vertical dotted line therethrough illustrates symbolically that portion ascribed to the local computer. The complete local computer is that portion of FIG. 1 necessary only to compute the required power generation by each power source of the local power transmission system.

Referring once more to Phil. 1, the signal delivered by each one of output terminals 55 is applied to a respective one of potentiometers 6t 61, 62, 63 and 64. The movable arms of each of potentiometers 6$64 are mechanically ganged together and positioned by a servomotor 66. Each of potentiometers 69-64 is adapted to multiply the value of the signal received from the corresponding one of terminals 55 by the common incremental cost of received power of the Southern system; name- 1y, A Thus, the like positions of the movable arms of p tentiorneters 6@64- correspond to the value of A Each output signal delivered by the movable arm of one of potentiometers 6944 represents the incremental cost of transmission losses in the local system (i.e., dollars per megawatt-hour) for the corresponding generating station, the cost being based on the common incremental cost of power, A of the Southern system. Therefore,

these incremental transmission losses are charged at the common incremental cost of received power of the Southern system. The movable arms of potentiometers 60454 are connected respectively to differential amplifiers 68, 69, ill, '71, and '72 and apply thereto the corresponding incremental cost of transmission loss signal. The value of each of these output signals is given by the last term of the corresponding one of Equations 10.

A potentiometer 74-, whose movable arm is also mechanically ganged with the movable arms of potentiometers 66PM delivers an output signal, the amplitude of which is proportional only to the position of the movable arm. A fixed direct voltage is coupled to the input terminal 75 of potentiometer 74. Since the position of the movable arm of potentiometer '74 corresponds to a the output signal delivered by the movable arm of potentiometer "id represents this common incremental cost of received power; that is, A The output signal of potentiorneter '74 is applied to each of differential amplifiers 68-72.

Each of differential amplifiers 68-72 is adapted to subtract the value of the output signal of the corresponding one of potentiometers fill-64 from the value of the output signal of potentiometer 74. The output signals of differential amplifiers 63 n. are given by the corresponding ones of Equations 16, and represent, therefore, the incremental cost of power generation for the corresponding generating station of the Southern system.

Th output signals of differential amplifiers 68-72 are applied to respective ones of station units 3tl34. Each of station units 3tl.Z- l responds to its input signal, which represents the incremental cost of power generation of the corresponding generating station, to deliver an output signal, as described heretofore, which represents the total power generation of the corresponding generating station.

A station unit of the type shown in FIG. 3 is useful in the apparatus of FIG. 1. The signal applied to an input terminal 86 of the station unit represents the incremental cost of power generation of the generating station represented thereby, this signal being generated in accordance with Equations 10. The signal at terminal 8% is applied to a multiplier Si. Devices for multiplying two signals are well known in the art and they may comprise, for example, the previously described potentiometer multipliers or one of the multipliers described in chapter 6 of the aforementioned Electronic Analog Computers, by Korn and Korn. Applied to an input terminal 82 is a signal representing the inverse of the fuel cost for the corresponding generating station. The fuel cost is a predetermined quantity and its inverse is usually manually set into the station unit, as by adjustment of a potentiometer, not shown. Dividing the incremental cost of power generation by the cost of fuel gives the incremental fuel rate. For example, if the signal applied to terminal 39 represents the incremental cost of power generation in units of dollars per megawatt-hour and the signal applied to terminal 82 represents the inverse of the fuel cost in terms of B.t.u.s per dollar, the output signal delivered by multiplier 1 represents B.t.u.s per mcgawatt-hour. For any given electrical power output, the incremental fuel rate is the ratio of the small change in fuel input to the corresponding small change in electrical power output. For steam generating stations, this incremental fuel rate uniquely determines the output power of a particular steam turbine-generator boiler unit. A curve of incremental fuel rate vs. power output may be termed an incremental fuel rate characteristic and is different for different generators. Such a curve is illustrated and explained in chapter 2 of the aforementioned Economic Operation of Power Systems, by L. K. Kirchmayer.

The output signal of multiplier 31 is applied to a plurality of function generators, such as function generators 84 and 85, each function generator representing the particular incremental fuel rate characteristic of a generator of the corresponding generating station. Function generators for providing an output voltage as a particular function of an input voltage are well known in the art and are described, for example, in chapter 6 of the aforementioned publication, Electronic Analog Computers, by Korn and Korn. Although the input signals to each of the function generators of a given station unit represent the same value of incremental fuel rate, the output signals provided by the function generators will usually represent different values of powers because of the differing characteristics of the generators.

The output signals of all function generators of the station unit are a d to a summing amplifier 86, the output signal of which is the output signal of the station unit and represents the total power generation of the generating station represented by that station unit.

Thus, as heretofore described, the local computer of FIG. 1 comprises a computing matrix for receiving input signals representing power from all sources for consumption by the local system load and in response to these input signals for providing output signals representing the incremental transmission losses in the local system for changes of power of each of the generating stations thereof; a multiplying means and a differential amplifier for each of these output signals to provide corresponding signals representing the incremental cost of power generation for each local generating station; and a station unit for each generating station responsive to the signals representing the incremental cost of power generation thereof for providing a signal representing the total power generated by that station. The local computer further comprises a summing amplifier responsive to all signals delivered by the station units for providing a signal representing the total local generation of the local system; a differential amplifier for providing a corresponding output signal whenever the required power consumption of the local system differs from the total power available both from the local generating stations and from interchange sources; and a servo system responsive to the output of the differential amplifier for adjusting the signal represent ing the common incremental cost of received power in the local system.

Thus, the local computer of this invention comprises a servomechanism of a type as described, for example, in fundamental form on page 64 of the aforementioned Electronic Analog Computers, by Korn and Korn, and in chapter 2 of B. K. Ledgerwood, Control Engineering Manual. This servomechanism is adapted to adjust the signals representing the total available power in the local system so that they remain equal to the input signals representing the required power consumption of the local system. Whenever the required power consumption signal applied to differential amplifier 37 deviates from the total of the power available signals applied to differential amplifier 37, an error signal representing the deviation is applied to servo amplifier 65. Servo amplifier 65 in turn supplies appropriate power to rotate servomotor 66 in a direction adapted to alter the local computer signals to reduce the magnitude of this deviation. Rotation of servomotor d6 alters the value of the local system a and, consequently, the incremental cost of power generation signal of each generating station. Each station unit responds to the corresponding altered incremental cost of power generation signal to change its total power output signal. The sum of the total power output signals of the station units as applied to differential amplifier 37 through summing amplifier 36 reduces the aforementioned deviation. The error signal delivered by differential amplifier 37 continues to diminish as the available power signals change to meet the power consumption requirement signals.

Although a specific form of local computer has been shown in FIG. 1, the embodiment of this invention is not limited to this particular type of computer, but instead any computing system may be used which responds to a signal representing the desired. total power to be consumed in the corresponding local system and a signal representing the algebraic value of power delivered to the local system by power sources external thereto for computing the required power generation by each of the power sources of the corresponding local system to satisfy the power consumption requirement and to deliver power to a predetermined location in the local system at a common incremental cost.

Concepts of Integrated System Computer For a better understanding of the embodiment of this invention to be described hereinafter, the concepts and theory thereof will now be explained. As has been previously stated, an integrated power transmission system is most economically operated when each local system of the integrated system receives power from each of the i other local systems thereof at the same incremental cost,

and when this incremental cost is the same as the incremental cost of power received from the local generating stations. An integrated system so operated is said to be, for purposes of this description, in economic dispatch. Thus, it is a function of the integrated system computer of this invention to calculate the interchange power requirements of each local power transmission system of the integrated system to maintain the integrated system in economic dispatch.

The particular embodiment of the integrated system computer to be described herein is adapted to calculate the necessary generation schedule to maintain in economic dispatch an integrated power transmission system formed by the generating stations of the Niagara Mohawk Power Corp. and the New York State Electric and Gas Corp. A simplified map of the integrated system is shown in FIG. 4. The Niagara Mohawk Power Corp. comprises three local power transmission systems designated respectively as the Western, Central and Eastern systems. The New York State Electric and Gas Corp. is located geographically south of the Niagara Mohawk Power Corp. and is designated, for simplicity, as the Southern local power transmission system. The integrated system of FIG. 4, therefore, comprises four local power transmission systems interconnected to exchange power.

The individual tie-lines, or transmission paths, of the integrated system are numbered for purposes of identification. Thus, tie-line 1 directly connects together the Western and Central systems, tie-line 2 directly connects together the Western and Southern systems, tie-lines 3, 4, 5 and 6 directly connect together the Southern and Central systems, and tie-line '7 directly connects together the Eastern and Central systems.

The integrated system of FIG. 4 is adapted to exchange power with transmission systems external thereto. Transmission systems external to the integrated system are designated as foreign systems X X X and X Power exchange with these foreign systems is not directly calculated by the integrated system computer, but the computer does provide pertinent cost information required for the determination of the economical foreign power exchange. Therefore, these foreign systems are not considered part of the integrated system. Tie-lines are also shown for interconnecting the integrated system with the foreign systems. For example, tie-line 8 directly connects the Eastern system with foreign system X and tie-line 15 directly connects the Western system with foreign system X One other tie-line, which plays a significant part in maintaining the integrated system in economic dispatch, is also shown in FIG. 4; that is, tie-line 17, which directly connects foreign systems X and X Consider, now, an integrated power transmission system comprising it local power transmission systems. For each local system it is necessary that the load requirements thereof be satisfied. The equation relating the required and available powers of a local system is the constraining equation. The n constraining equations for the integrated system are where i l/ represents the constraining equation of the k local system,

P represents the load requirements of the k local system,

P represents the transmission losses in the k local system,

P represents the net, or algebraic, value of interchange power received by the k local system from the other n-1 local systems of the integrated system and from the foreign systems, and

21 cm represents the total generation required of all i generating stations of the k local system.

In Equations 12 only nl of the P terms are independent variables and the remaining P term is dependent, so that any one of the P terms may be expressed as a fun"- tion of the other nl terms, as follows In Ia Ib 1,n 1

The total cost of power generation of the system is a function of the power delivered by all generating units of all local systems and may be expressed as Total Cost (F Ft +Ft where F represents the total cost of the generation of all generatin stations of the k local system.

It is known that where such a function F of a plurality of variables, which are connected by a number of distinct constraining equations, is to be minimized, the values of the variables which render this function a minimum may be determined by equating to zero the partial derivatives of the auxiliary function =FT+raa+rna+ int 1,.

wherein A A A are regarded as constants. This is shown, for example, in the aforementioned Mathematical Analysis" by Goursat and Hedrick.

Equating the partial derivatives of Equation 15 to zero, there result Substituting the expressions for of Equations 12 in Equations 16, there result,

Substituting the expression for P of Equation 13 in the last of Equations 12, and then substituting the expressions for b of Equations 12 in Equations 17, there result,

OP 1 for example, represents the incremental cost of Gei power generation for the 11 generating station of the (1 local system,

P for example, represents the ratio of change in a GM transmission losses in the a local system to change in power of the i generating station thereof when delivering an increment of power from this 11 generating station to the hypothetical load center of the a local system,

i for example, represents the ratio of change in 5 Get transmission losses in the b local system to change in power of the i generating station of the a local system when delivering an increment of power from the i generating station of the a local system to the hypothetical load center of the a local systern, and

M, for example, may be interpreted as the incremental cost of power received by the load center of the a local system,

i for example represents the ratio of change in a transmission losses in the a, local system to change in interchange power received by the a local system when the a local system receives an additional increment of interchange power, and i for example, represents the ratio of change in h transmission losses in the b local system to change in interchange power received by the a local system, when the a local system receives an addi tional increment of interchange power.

In developing Equations 19, P has been treated as the dependent variable, as in Equations 13.

It is to be noted that the first of Equations 18 represents, in shorthand form, a plurality of equations, there being one of these equations for each generating station of the a local system. Similarly, the number of equations represented by the second and each remaining equation of Equations 18 corresponds to the number of generating stations in the corresponding local system. For example, if the b local system has six generating stations, the second of Equations 18 would actually represent six equations, one for each generating station of the b local system.

Although rigorous Equations 18 may be implemented by an integrated system computer, the cost and complexity of such a computer may be considerably reduced by dropping quantities in Equations 18 that are very small compared to other terms therein. It may be shown, for example, in the first of Equations 18 that the sum of the terms, such as Lu. Gai

1ml oP are given by Equation 8a.

Equations 20 are, therefore, similar in form to Equaare given by Equation 8a. tions 6, which describe the operation'o-f the local computer of FlG. 1. Therefore, an additional advantage of employing simplified Equations 20 in the integrated system computer is that each local computer portion thereof need receive no signals from other local computers to compute the local system generation requirements. For example, a local computer for the a local system, as described by the first of Equations 18 must receive signals representing incremental transmission losses [in each of the b to n local systems from the corresponding local computers, whereas a local computer described by the first of Equations 20 need receive signals representing incremental transmission losses from no other local computers.

Equations 20 are termed the iutra-area coordination equations of the integrated system. Each intra-area equation is concerned with the incremental transmission losses and incremental costs of power of only the corresponding local system. Equations 19 are termed the inter-area coordinated equations for the integrated system. Each interarea equation is concerned with incremental transmission losses and incremental costs of power throughout the integrated system.

Equations 19 may be rewritten as Inter-area coordination Equations 21 may be interpreted as follows. Assume, on one hand, that each local system, except the n local system, receives an additional increment of interchange power. All of these increments of interchange power must, therefore, be supplied by the n local system. Under these conditions, Equations 21 specify that for each local system to receive interchange power at the same incremental cost as it is receiving power from its local generating stations, the incremental cost of this interchange power as transmitted at the n local system must be the same as the incremental cost of power received by the load center of the n local system from its local generating stations. Assume, on the other hand, that each local system, except the n local system, transmits an additional increment of interchange power. The n local systemmust, therefore, receive allot these increments of interchange power. Under these conditions, Equations 21 specify that the incremental costs of interchange power received at the n local system must be the same for all of the other local systems and that this common incremental cost of interchange its local generating stations.

Equations 21 may be rewritten as follows hraic value of power delivered to the cor-responding local system by the other local systems (i.e., P of Equations 12). In response to this pair of input signals, and to the computed common incremental cost of power, 7t, of

Inter-area coordination Equations 22 may be interpreted as follows. The respective incremental costs of power delivered to the boundary of the n local system for an incremental transfer of interchange power to each of the other local systems must be equal to the respective required costs of power at this boundary for each of the other local systems to receive this increment of interchange power at the respective load centers thereof at the same incremental cost as power is received from the respective local generating stations. Again, a converse interpretation may he placed on Equations 22. That is, each of the local systems, other than the n local system, must deliver an increment of interchange power to the boundary of the n local system at an incremental cost equal to the corresponding required incrementm cost of power at this boundary for the n local system to receive this increment of power at its load center at the same incremental cost as power is received from its local generating stations. Both groups of Equations 21 and 22 represent the theory of operation of the integrated systern computer of this invention.

For purposes of clarity in the succeeding description, the aforementioned n local system is termed the reference system and the remaining rz1 local systems are termed peripheral systems. Equations 21 and 22 specify that in the integrated system computer a pair of incremental costs of interchange power are determined at a predetermined location in the integrated system for each of the peripheral systems for transfer of an increment of power from the corresponding peripheral system to the reference system or for transfer of an increment of power from the reference system to the corresponding peripheral system. One value of each of these pairs is based on the incremental cost of power received at its load center by the corresponding peripheral system from the generating stations thereof and the other value of each pair is based on the incremental cost of power received at its load center by the reference system from the generating stations thereof.

FIGURE 5 illustrates one concept of operation of the integrated system computer of this invention. FIGURE 5 is not to be considered as a block diagram of the computer itself, but is only employed as an aid in the understanding of the operation of the embodiment to be described hereinafter. Single-area computing apparatuses 9t 91, 92 and 93, which are similar to the apparatus shown in FIG. 1, are provided for the respective Western, Southern, Eastern and Central local power transmission systems shown in FIG. 4. The Central local system is established as the reference system in the integrated systern computer. Each of apparatuses 9%i-93 receives a pair of input signals, one of these input signals representing the desired total power to be consumed in the corresponding local system (i.e., P +P of Equations 12) and the other of these input signals representing the algethe corresponding local system, each single-area conputing apparatus of a peripheral system delivers a signal representing the incremental cost of power at the boundary of the reference system for transfer of an increment of power from the corresponding peripheral system to the reference system, this incremental cost being based on said value of A. l' he reference system sin le-area computing apparatus 93 provides a signal for each peripheral system rep-resenting the required incremental cost of power at the boundary of the reference system for transfer of an increment of power from the corresponding peripheral system to the reference system, this incremental cost being based on the value of A computed by apparatus 93.

The incremental cost of power signal delivered by apparatus 9t and the corresponding signal delivered by apparatus are applied to a differential amplifier and signal generator 95. Differential amplifier and signal generator 95 is adapted to provide an output signal representing the value of interchange power, P delivered to the Western local system and in response to any difference in the pair of signals received thereby to alter the value of this output signal in such a sense as to tend to reduce the magnitude of this difference. The signal delivered by apparatus W represents the left side of an Equation 22 and the corresponding signal delivered by apparatus 93 represents the right side of this equation. If these two signals are unequal, this equation is not satisfied. Differential amplifier and signal generator 95 thereupon readjusts the interchange power signal, P; in such a sense as to tend .to satisfy this equation. in a similar manner, differential amplifier and signal generators 6 and 97 receive pairs of corresponding incremental cost of power signals and in response thereto deliver signals representing the required value of interchange power for the respective Southern and Eastern local systems. All output signals of differential amplifier and signal generators S55, 96 and 97 are applied to a summing amplifier 953, the inverted output signal of which represents the value of interchange power received by the Central local system, in accordance with Equation 13.

In the integrated system computer concept of PEG. 5, each incremental cost of power signal may, alternatively, represent the incremental cost of power at the boundary of the reference system for transfer of an increment of power from the reference system to the corresponding peripheral system. The computer of FIG. 5, in accordance with Equations 22, is also adapted to compare the incremental costs of interchange power at other desired locations in the integrated system and is not limited in its computation to the reference system boundary. In fact, the location chosen may be the reference local system load center, in accordance with Equations 21.

Thus, each peripheral system computing apparatus of thi invention provides an output signal representing the incremental cost of power at a corresponding predetermined location in the integrated system for a transfer of power between the corresponding peripheral system and I the reference system, this incremental cost being based on the common incremental cost of power at the peripheral system. The reference system computing apparatus prcvides an output signal for each peripheral system representing the incremental cost of power at the corresponding predetermined location for a transfer of power between the corresponding peripheral system and the reference system, this incremental cost being based on the common incremental cost of power at the reference systern.

In its more general concept, the computing apparatus of this invention provides a pair of signals for each of 11-1 different pairs of local systems of an integrated system comprising 11 local systems. The two signals of each signal pair represent respectively the incremental costs of power at a corresponding predetermined location for a hypothetical transfer of an increment of power between the corresponding pair of local systems, based on the common incremental cost of power at each of the two corresponding local systems. In response to the differences between the two signals of each of the n-1 signal pairs, n signals are derivexi, which represent respectively the value of interchange power received by the n local systems.

Consider once again FIG. 4. Power may be transferred between any two local systems of the integrated system either over the tie-lines directly interconnecting the two systems or through other local systems. For example, power may be exchanged between the Western and Central local systems directly over tie-line 1. Power may also be transferred between the Western and Central systems through the Southern local system by routing power over tie-line 2 and then over tie-lines 3-6. Similarly, power may be exchanged between the Central and Southern local systems directly over tie lines 3-6 or through the Western local system ower tie-lines 1 and 2. Power transferred from one local system to another through a third local system is termed wheeled power. A change in wheeled power results in a change of transiission losses in the system through which the power is wheeled. For example, when power wheeled through the Western local system is increased during an increase in power exchanged between the Central and Southern local systems there is a corresponding change of power on tie-lines 1 and 2. This change results in an increase in wheeling losses incurred in the Western local system. Such changes in losses in a system through which power is wheeled are referred to as incremental wheeling losses.

Therefore, whenever a variation in interchange power occurs, the computation of the incremental cost of power at the reference system boundary must take into consideration the incremental cost of these wheeling losses. Thus, all terms on the left-hand side of Equations 22 which involve two areas, are terms representing the incremental cost of wheeling losses. For example, the term represents the incremental cost of wheeling losses in the 12 local system for a change in interchange power of the n local system. The latter portion of this term represents the incremental wheeling losses in the b local system. Although computation involving the incremental cost of wheeling losses is not shown in FIG. 5, for purposes of simplicity, the embodiment of this invention provides for inclusion of wheeling loss terms.

Wheeling power is not confined to the integrated system, itself, but power may be transferred between two local systems of the integrated system through at least one of the foreign systems connected to the integrated system. For example, power may be transferred between the Western and Central systems through foreign system X In this example, since the foreign system is not a part of the integrated system, its incremental cost of wheeling losses is not considered by the integrated system computer. However, where power is transferred between the Southern and Central systems through the cascaded foreign system X foreign system ,X and the Eastern system, the incremental cost of wheeling losses through the Eastern local system is considered by the integrated system computer.

Individual T {e -Line Considerations Inter-area coordination Equations 21 and 22 treat each value of interchange power in its entirety. Thus, the typical term La 8P represents the incremental losses in the a local system for a variation in total interchange power received by the a local system. However, the interchange power received by a local system is transmitted thereto on a plurality of tie-lines. For example, the Western system receives interchange power directly from the Central and Southern systems on respective tie-lines 1 and 2, and from foreign systems X and X; on respective tie-lines l4 and 15.

A tie-line in an integrated system is adapted to transfer power directly between the pair of local systems connected thereto and to transfer wheeling power through either one of this pair of local systems. When an additional increment of interchange power is received by a system, this increment is distributed among the tie-lines connected thereto in accordance with the relative self and mutual impedances of the tie-lines and in accordance with the amount of power already flowing over each tieline. The resulting incremental transmission losses associated with each tie-line contribute toward the total incremental losses in a local system for, variation in interchange power. In a similar manner, the total incremental Wheeling losses in a local system depend on the distribution of wheeled power over the group of tie-lines transmitting the wheeled power into the local system and over the group of tie-lines transmitting the wheeled power out of the local system. Therefore, to determine the values of the incremental loss terms of Equations 21 and 22, the individual tie-line incremental loss contributions must be determined and then properly combined.

Since the incremental power losses in a local system are a function of the individual powers transmitted on each tie-line connected thereto, and since each of these individual tie-line powers is a function of the total interchange power received by a local system, each incremental loss term of the inter-area coordination Equations 21 and 22 may be expressed as 5 rs Li 'ra m y Tjy n where Z {3 represents the ratio of the change in power a 1k transmitted over tie-line y into system j to the increment of total interchange power delivered to the is local system.

r 41 Substituting Equation 23 into Equations 22, there result various combinations of calculated incremental power losses in the corresponding local system for incremental The terms B Li Ti are determined by partially differentiating Equation 7 with respect to the terms representing the particular tieline powers,

where The values of E are determined by the same methods by which the values or" B in Equation 7 are determined.

The values of the terms fi may be determined by conventional circuit analysis methods, wherein the change in power of one of a plurality of transmission paths in a network may be determined from network parameters and a knowledge of the change of total power over an aggregate of these transmission paths. Specific methods of computation are indicated in Analysis of Losses in Loop-Interconnected Systems, A. F. Glimn, L. K. Kirchrnayer, G. W. Stagg, AlEE Trans, vol. 72, part III, 1953, pp. 796-807; and in Improved Method of Interconnecting Transmission Loss Formulas, A. F. Glimn, L. K. Kirchmayer, J. J. Skiles, AIEE Trans. Paper 5,8 5l3, 1958.

Integrated System Computer The integrated system computer shown in FIGS. 64 and 6b is adapted to calculate the necessary generation schedule to maintain in economic dispatch the integrated power transmission system shown in FIG. 4. Singlearea computing apparatuses 1431, 102, 193 and 1%, each of the type shown in PEG. 1, are provided respectively for the Western, Southern, Central and Eastern local systems of the integrated system. In this computer, the Centnal local system has veen designated as a reference system. Although a particular local system has been selected as a reference system for purposes of computation, where it is desired to employ a reference system in the computation, any other local system could have been selected and the computer designed in accordance with such selection.

Each one of single-area computing apparatuses 19% 164 comprises a local computer, which has previously been described in detail. In addition to the local computer portion thereof, each of these single-area computing apparatuses comprises additional means for providing changes of power on each of the tie-lines connected to the local system. Each single-area computing apparatus receives a plurality of signals representing respectively the power transmitted on each tie-line connected to the corresponding local system; a signal representing the desired total power to be consumed in the corresponding local system; and a signal representing the algebraic value of interchange power delivered to the corresponding local system. Each single-area computing apparatus delivers a plurality of signals representing respectively the total power supplied by each generating station of the corresponding local system; a plurality of signals representing the cost of transmission losses in the corresponding local system for an incremental change in total interchange power received thereby, or in interchange power transmitted from the corresponding local system to a foreign system, these signals hereinafter being termed cost weighting signals; and a plurality of signals representing the incremental cost of wheeling losses for transferring wheeled power through the corresponding local system, these signals hereinafter being termed wheeling cost signals.

Within each block that symbolically represents a singlearea computing apparatus, in FIG. 6, a received signal representing a tie-line transmitted power is identified by the notation P where y identifies the tie-line number; a delivered signal representing a generating station power output is identified by the notation P where ji signifies the i generating station of the j local system; adelivered cost weighting signal is identified by the notation I J (CW) where I and J represent the two systems exchanging power for which the corresponding incremental cost of losses is computed; and a delivered wheeling cost signal is identified by the notation 1 -4 (WH), where J and 3 represent the two systems exchanging power for which the corresponding incremental .cost of wheeling losses is computed.

In the preceding discussion of the concept of the integrated system computer, shown in FIG. 5, it was pointed out that the signalsrepresenting the values of interchange power required to maintain the integrated system in economic dispatch are obtained by comparing signal pairs representing the incremental cost of power at predetermined locations in the integrated system for transfer of power between the peripheral systems and the reference system. These incremental cost of power signals, described by each side of Equations 22, are computed from three types of signals, a first type of signal, A, representing the common incremental cost of received power in a local power transmission system; a second type of signal,

representing the incremental transmission losses in a local system for a change in power wheeled therethrough. Details of the theory and procedures by which the second and third signal types may be obtained are shown in Equations 23 and 24. That portion of the single-area computing apparatus of FIG. 1 comprising the portion of matrix 40 to the right of the vertical dotted line; potentiometers 106, 107, 168, 1639, 110 and 111; and incremental cost of loss matrix 114 is adapted to provide these second and third signal types.

In the incremental transmission loss matrix 40 of FIG. 1, shown in detail in FIG. 2, a combining lead 511 is provided for each tie-line connected to the corresponding local system, in addition to the combining leads 5% provided for the local generating stations. Six tie-lines are connected to the Southern system so that six corresponding combining leads 50 must be provided. Each input signal to the matrix is multiplied by a corresponding one of potentiometers 47 for each tie-line combining lead 59. Since eleven input signals are supplied to the Southern system matrix 40, eleven potentiometers 47 are coupled to each tie-line combining lead 51). Each potentiometer 47 so coupled to a tie-line combining lead 51? is adjusted to represent the corresponding magnitude of B of Equation 25. The corresponding switch 48 is set in accordance with the polarity designation of B For matrix 41), the Values of the subscript m of E range from 1 to 11, depending on the one of the eleven input signals to be multiplied by the corresponding potentiometer 47; S is substituted for the subscript to represent the Southern local system; and the subscript y bears the values 2, 3, 4, 5, 6, and 16, depending on the particular tieline corresponding to the combining lead 56) to which the potentiometer 47 is coupled.

The eleven signal applied to each combining lead 5% are algebraically added by the corresponding one of summing amplifiers 54, and a signal representing the summation is delivered at a corresponding output terminal 56. Each signal delivered at a terminal 56, therefore, represents the incremental transmission losses in the Southern system for the corresponding tie-line,

B Ls DPTSY Therefore, these output signals of potentiometers 107- 111 represent respectively the costs of transmission losses in the Southern local system for an incremental change in power delivered to the Southern local system on each of tie-lines 2, 3, 4, 5, 6 and 16.

The output signals of potentioineters 106-111 are applied to the input terminals of incremental cost of loss matrix 114. Matrix 114- is similar in construction to the matrix of FIG. 2. Matrix 11 1, therefore, is adapted to multiply each incoming signal by a plurality of discrete values, and to provide various combinations of the resulting products as the aforementioned cost weighting and wheeling cost signals.

An example of one of the output signals delivered by matrix 114 is the cost wei hting signal representing the cost of transmission losses in the Southern system for an incremental change of total interchange power received by the Southern system. The value of this incremental cost of losses is given by where represents the cost of losses in the Southern system for an incremental change in total interpower delivered to the Southern system.

Equation 26 is seen to be a particular term of the interarea coordination Equations 24.

Equation 26 may be expanded to illustrate how matrix 114 operates on the signals received thereby to generate the incremental cost of loss signal represented by Equation 26, as follows 'rsiu The signal represented by Equation 27 is provided by a summing amplifier 54 terminating one of the combining leads 54 of matrix 114, this combining lead 50 receiving the six signals represented by the right-hand portion of Equation 27. From Equation 27 it is seen that each signal received by matrix 114 is multiplied by gi These values of fi are provided by adjusting the corresponding potentiometers 47 and switches 48 of the matrix.

An example of a wheeling cost signal delivered by matrix 114 is that which represents the cost of transmis sion losses in the Southern local system for an incremental change of total interchange power received by the Western system, and is given by The six potentiometers 47 coupled to the combining lead 50 which delivers the six signals represented by the righthand portion of Equation 28 are adjusted to represent respectively the values of fi shown therein.

Another example of a wheeling cost signal delivered by matrix 114 is that which represents the cost of transmission losses in the Southern system for an incremental change in power delivered by the Central system to a predetermined location in foreign system X and is given by In all, seven signals representing the incremental cost of losses in the Southern system for respective changes of power exchanged between local systems or between a foreign system and a local system are delivered by matrix 114. The settings of all potentiometers in matrix 114 

1. APPARATUS FOR COMPUTING THE ECONOMIC DISTRIBUTION OF POWER OF A PLURALITY OF LOCAL POWER TRANSMISSION SYSTEMS INTERCONNECTED TO FORM AN INTEGRATED POWER TRANSMISSION SYSTEM, WHEREIN EACH OF SAID LOCAL SYSTEMS IS CONNECTED BY AT LEAST ONE TRANSMISSION PATH TO EXCHANGE POWER WITH AT LEAST ONE OTHER OF SAID LOCAL SYSTEMS, AND WHEREIN ONE OF SAID LOCAL SYSTEMS IS DESIGNATED AS A REFERENCE SYSTEM AND THE REMAINDER OF SAID LOCAL SYSTEMS ARE DESIGNATED AS PERIPHERAL SYSTEMS, COMPRISING; MEANS FOR PROVIDING A PLURALITY OF FIRST SIGNALS REPRESENTING RESPECTIVELY THE INCREMENTAL COST OF POWER OF A CORRESPONDING ONE OF SAID LOCAL SYSTEMS; A FIRST COMPUTING MEANS FOR EACH OF SAID PERIPHERAL SYSTEMS RESPONSIVE TO THE CORRESPONDING FIRST SIGNAL FOR COMPUTING FOR EACH OF SAID TRANSMISSION PATHS CONNECTED TO SAID CORRESPONDING PERIPHERAL SYSTEM THE INCREMENTAL COST OF TRANSMISSION LOSSES IN SAID PERIPHERAL SYSTEM FOR TRANSFER OF POWER ON SAID TRANSMISSION PATH, SAID COMPUTATION BEING BASED ON THE INCREMENTAL COST OF POWER REPRESENTED BY SAID CORRESPONDING FIRST SIGNAL, SAID FIRST COMPUTING MEANS PROVIDING A PLURALITY OF SECOND SIGNALS REPRESENTING RESPECTIVELY THE INCREMENTAL COST OF LOSSES COMPUTED THEREBY; A SECOND COMPUTING MEANS FOR EACH OF SAID PERIPHERAL SYSTEMS RESPONSIVE TO THE CORRESPONDING FIRST SIGNAL AND TO THE CORRESPONDING PLURALITY OF SECOND SIGNALS FOR COMPUTING THE CORRESPONDING INCREMENTAL COST OF POWER AT A CORRESPONDING PREDETERMINED LOCATION IN SAID INTEGRATED SYSTEM FOR A TRANSFER OF POWER BETWEEN THE CORRESPONDING PERIPHERAL SYSTEM AND SAID REFERENCE SYSTEM, SAID SECOND COMPUTING MEANS PROVIDING A THIRD SIGNAL REPRESENTING THE INCREMENTAL COST OF POWER COMPUTED THEREBY; A THIRD COMPUTING MEANS RESPONSIVE TO THE ONE OF SAID FIRST SIGNALS REPRESENTING THE INCREMENTAL COST OF POWER OF SAID REFERENCE SYSTEM FOR COMPUTING FOR EACH OF SAID TRANSMISSION PATHS CONNECTED TO SAID REFERENCE SYSTEM THE INCREMENTAL COST OF TRANSMISSION LOSSES IN SAID REFERENCE SYSTEM FOR TRANS- 